Question

If $\tan A-\tan B=x$ and $\cot B-\cot A=y$ then $\cot (A-B)=$

• A. $\frac{1}{y}-\frac{1}{x}$
• B. $\frac{1}{x}-\frac{1}{y}$
• C. $\frac{1}{x}+\frac{1}{y}$
• D. None of these

Answer 1

Answer: (C)

$\frac{1}{x}+\frac{1}{y}$

Solution:

We have, $\tan A-\tan B=x,\cot B-\cot A=y$

To find:

$\cot (A-B)=?$

Solution:

$\tan A-\tan B=x$

or, $\frac{1}{\cot A}-\frac{1}{\cot B}=x$

or, $\frac{\cot B-\cot A}{\cot A\cot B}=x$

or, $\cot A\cot B=\frac{y}{x}$

Now, $\cot (A-B)=\frac{\cot A\cot B+1}{\cot B-\cot A}$

$=\frac{\frac{y}{x}+1}{y}$ [From above]

$=\frac{1}{x}+\frac{1}{y}$

Hence, C is the correct option.